By Sadao Kawamura; Mikhail Svinin
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Its form gives us a hint: it would be convenient to augment the conﬁguration space with the path parameter q∗ . Doing so, we obtain the augmented conﬁguration space C ∗ , a ¯ ≡ (q, q∗ ). Equation (6) can be rewritten subspace of n+1 , with elements q in compact form, as: 1 In case of a mixed-joint structure (with rotational and translational joints), uniformity can be ensured with proper scaling. 12 Dragomir N. Nenchev J (¯ q )¯q˙ = 0 (7) where J (¯ q ) ≡ J (q) −t∗ (q∗ ) is called the column-augmented Jacobian.
The kinematic singularity at the intersection is a regular point singularity. Simulation data are shown in Fig. 2. The x − y graph in the lower left part of the ﬁgure shows the workspace boundary (the full circle) and the path described by the end-tip (the circular arc). Note that the q∗ speed graph is actually the graph of determinant det J . After about 3 s, change of its sign is observed which indicates motion through the kinematic singularity at the intersection point between workspace boundary and end-tip arc.
The self-motion manifold can then be characterized by the invariant arc lenght λ, called also natural parameter. λ is determined uniquely up to an additive constant, via: ¯ . λ˙ = n (14) λ˙ will be referred to as the natural speed along the self-motion manifold. From Eq. (8) we obtain: ˆ ¯q˙ = λ˙ n ¯ (¯ q) (15) ˆ is the tangent vector of unit length at q ¯ , and the constant b has been where n ¯ set to one, without loss of generality. Deﬁnition (Natural motion) Manipulator motion with generalized velocities in proportion to the natural speed λ˙ is called natural motion.
Advances in Robot Control by Sadao Kawamura; Mikhail Svinin